Dynamic Causal Modelling Advanced Topics SPM Course (fMRI), May 2015 Peter Zeidman Wellcome Trust Centre for Neuroimaging University College London

Dynamic Causal ModellingAdvanced Topics

SPM Course (fMRI), May 2015

Peter Zeidman

Wellcome Trust Centre for Neuroimaging

University College London

The system of interest

Stimulus from Buchel and Friston, 1997Brain by Dierk Schaefer, Flickr, CC 2.0

Experimental Stimulus (Hidden) Neural Activity Observations (BOLD)

time

Vector y

BO

LD

?off

on

time

Vector u

https://creativecommons.org/licenses/by/2.0/

DCM Framework

Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0

Experimental Stimulus (u)

Observations (y)

z = f(z,u,θn).

How brain activity z

changes over time

y = g(z, θh)

What we would see in the scanner, y, given the

neural model?

Neural Model Observation Model


DCM Framework


Observations (y)Neural Model Observation Model

Model Inversion(Variational EM)

1. Gives parameter estimates:Given our observations y, and stimuli u, what parameters θ make the

model best fit the data?

2. Gives an approximation to the log model evidence:

Free energy = accuracy - complexity

Contents

• Creating models– Preparing data for DCM

• Inference over models– Fixed Effects– Random Effects

• Inference over parameters– Bayesian Model Averaging

• Example

• DCM Extensions

PREPARING DATA

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Choosing Regions of Interest

We generally start with SPM results

Main effects → driving inputsInteractions → modulatory inputs FFA Amy

Face

Valence

From a factorial design:

A factorial design translates easily to DCM

A (fictitious!) example of a 2x2 design:

Factor 1:Stimulus (face or inverted face)

Factor 2:Valence (neutral or angry)

Main effect of face: FFA

Interaction of Stimulus x Valence: Amygdala

+

ROI Options

1. A sphere with given radius

Positioned at the group peak

Allowed to vary for each subject, within a radius of the group peak

or

2. An anatomical mask

Pre-processing

1. Regress out nuisance effects (anything not specified in the ‘effects of interest f-contrast’)

2. Remove confounds such as low frequency drift

3. Summarise the ROI by performing PCA and retaining the first component

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1st eigenvariate: test

time \{seconds\}

230 voxels in VOI from mask VOI_test_mask.niiVariance: 81.66%

New in SPM12: VOI_xx_eigen.nii(When using the batch only)

Interim Summary: Preparing Data

• DCM helps us to explain the coupling between regions showing an experimental effect

• We can choose our Regions of Interest (ROIs) from any series of contrasts in our GLM

• We use Principle Components Analysis (PCA) to summarise the voxels in each ROI as a single timeseries

Contents




• Example

• DCM Extensions

INFERENCE OVER MODELS





Model 1:

Model 2:

Model comparison: Which model best explains my observed data?

Bayes Factor

𝐵 𝑗𝑖=𝑝 (𝑦∨𝑚= 𝑗)𝑝 (𝑦∨𝑚=𝑖)

How good is model versus model in one subject?

log 𝐵 𝑗𝑖=log𝑝 (𝑦∨𝑚= 𝑗 )− log𝑝 (𝑦∨𝑚=𝑖)≅ 𝐹 𝑗−𝐹 𝑖

In practice we subtract the log model evidence:

𝐺𝐵 𝐹 𝑗𝑖=∏𝑘

𝐵𝑖𝑗𝑘

At the group level, over K subjects:

Bayes Factor - Interpretation

A log Bayes factor of 3 is termed “strong evidence”.

exp(3) = 20 x stronger evidence for model the than model

Kass and Raftery, JASA, 1995.

We can convert the Bayes Factor to aposterior probability using a sigmoid function

BF of 3 = 95% probability

Will Penny

Fixed Effects Bayesian Model Selection (BMS)

Assumption: Subjects’ data arose from the same underlying model

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Log-

evid

ence

(rel

ative

)

Models

Bayesian Model Selection: FFX

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el P

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rior P

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bility

Bayesian Model Selection: FFX

Models

Random Effects Bayesian Model Selection (BMS)

11 out of 12 subjects favour model 2But… GBF = 15 in favour of model 1

Stephan et al. 2009, NeuroImage



SPM estimates a hierarchical model with variables:

The frequency of each model in the group The model assigned to subject The data for subject

𝐸 [𝑟2|𝑌 ]=0.84

𝑝 (𝑟2>𝑟1|𝑌 )=0.99

Expected probability of model 2

Outputs:

Exceedance probability of model 2


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Models

Bayesian Model Selection: RFX

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Models


Assumption: Subjects’ data arose fromdifferent models



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Prot

ecte

d Ex

ceed

ance

Pro

babi

lity

Models

Prob of Equal Model Frequencies (BOR) = 0.45

Rigoux et al. 2014, NeuroImage

Exceedance probability (xp) assumes each model has a different frequency in the group. It is over-confident.

Bayesian Omnibus Risk (BOR)The probability that all models have the same frequency in the group:

The xp is adjusted using the BOR to give the protected exceedance probability (pxp).

Family Inference

Rather than having one hypothesis per model, we can group models into families.

E.g. we have 9 models which all have a bottom-up connection and 5 models which all have a top-down connection.

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el E

xpec

ted

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abili

tyModels


Family Inference

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ily E

xpec

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abilit

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Families


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Families


Interim Summary: Inference over Models

• We embody each of our hypotheses as a model or as a family of models.

• We can compare models or families using a fixed effects analysis, only if we believe that all subjects have the same underlying model

• Otherwise we use a random effects analysis and report the protected exceedance probability and the Bayesian Omnibus Risk.

Contents




• Example

• DCM Extensions

INFERENCE OVER PARAMETERS

Parameter Estimates

The estimated parameters for a single model can be found in each DCM.mat file

Inspect via the GUI Inspect via Matlab

Estimated parameter means: DCM.Ep

Estimated parameter variance: DCM.Cp

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el E

xpec

ted

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abili

ty

Models


Bayesian Model Averaging

Bottom-up Top-down00.10.20.30.40.50.60.70.80.9

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ily E

xcee

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obab

ility

Families

What are the posterior parameter estimates for the winning family?

𝑝 (𝜃|𝑦 )=∑𝑚

𝑝 (𝜃 ,𝑚∨𝑦)

¿∑𝑚

𝑝 (𝜃|𝑚 , 𝑦 )𝑝 (𝑚∨𝑦 )

SPM calculates a weighted average of the parameters over models:

To give a posterior distribution for each connection:

Region 1 to 2

Contents




• Example

• DCM Extensions

EXAMPLE

Reading > fixation (29 controls)Lesion (Patient AH)

1. Extracted regions of interest. Spheres placed at the peak SPM coordinates from two contrasts:

A. Reading in patient > controls B. Reading in controls

2. Asked which region should receive the driving input

3. Performed fixed effects BMS in the patient and random effects BMS in the controls, and applied Bayesian Model Averaging.

Seghier et al., Neuropsychologia, 2012

Key:ControlsPatient

Bayesian Model Averaging

Seghier et al., Neuropsychologia, 2012

Interim Summary: Example

• When we don’t know how to model something, we can ‘ask the data’ using a model comparison

• Bayesian Model Averaging (BMA) lets us compare connections across patients and controls

• Posterior parameter estimates can be used as summary statistics for further analyses

DCM EXTENSIONS

DCM Framework

Stimulus from Buchel and Friston, 1997Figure 3 from Friston et al., Neuroimage, 2003Brain by Dierk Schaefer, Flickr, CC 2.0


Observations (y)

z = f(z,u,θn).

How brain activity z

changes over time

y = g(z, θh)

What we would see in the scanner, y, given the

neural model?

Neural Model Observation Model


DCM Extensions

Non-Linear DCM Two-State DCM

driving input

modulation


Marreiros et al. 2008, NeuroImage

DCM Extensions

Stochastic DCM DCM for CSD

Li et al. 2011, NeuroImage

Friston et al. 2014, NeuroImage

DCM Extensions

Post-hoc DCM

Friston and Penny 2011, NeuroImage

Further Reading

The original DCM paper Friston et al. 2003, NeuroImage

Descriptive / tutorial papers

Role of General Systems Theory Stephan 2004, J Anatomy

DCM: Ten simple rules for the clinician Kahan et al. 2013, NeuroImage

Ten Simple Rules for DCM Stephan et al. 2010, NeuroImage

DCM Extensions

Two-state DCM Marreiros et al. 2008, NeuroImage

Non-linear DCM Stephan et al. 2008, NeuroImage

Stochastic DCM Li et al. 2011, NeuroImageFriston et al. 2011, NeuroImageDaunizeau et al. 2012, Front Comput Neurosci

Post-hoc DCM Friston and Penny, 2011, NeuroImageRosa and Friston, 2012, J Neuro Methods

A DCM for Resting State fMRI Friston et al., 2014, NeuroImage

EXTRAS

Approximates: The log model evidence:

Posterior over parameters:

The log model evidence is decomposed:

The difference between the true and approximate posterior

Free energy (Laplace approximation)

Accuracy Complexity-

Variational Bayes

The Free Energy

Accuracy Complexity-

Complexity

posterior-prior parameter means

Prior precisions

Occam’s factor

Volume of posterior parameters

Volume of prior parameters

(Terms for hyperparameters not shown)

Distance between prior and posterior means

DCM parameters = rate constants

dxax

dt 0( ) exp( )x t x at

The coupling parameter a thus describes the speed ofthe exponential change in x(t)

0

0

( ) 0.5

exp( )

x x

x a

Integration of a first-order linear differential equation gives anexponential function:

/2lna

00.5x

a/2ln

Coupling parameter a is inverselyproportional to the half life of z(t):

Practical Workshop

DCM – Attention to MotionParadigm

Parameters - blocks of 10 scans - 360 scans total- TR = 3.22 seconds

Stimuli 250 radially moving dots at 4.7 degrees/s

Pre-Scanning 5 x 30s trials with 5 speed changes (reducing to 1%)Task - detect change in radial velocity

Scanning (no speed changes)F A F N F A F N S ….

F - fixation S - observe static dots + photicN - observe moving dots + motionA - attend moving dots + attention

Attention to Motion in the visual system

Slide by Hanneke den Ouden

Results

Büchel & Friston 1997, Cereb. CortexBüchel et al. 1998, Brain

V5+

SPCV3A

Attention – No attention

- fixation only- observe static dots + photic V1

- observe moving dots + motion V5- task on moving dots + attention V5 + parietal cortex


Paradigm


V1

V5

SPC

Motion

Photic

Attention

V1

V5

SPC

Motion

PhoticAttention

Model 1attentional modulationof V1→V5: forward

Model 2attentional modulationof SPC→V5: backward

Bayesian model selection: Which model is optimal?

DCM: comparison of 2 models


DCM – GUI basic steps

1 – Extract the time series (from all regions of interest)

2 – Specify the model

3 – Estimate the model

4 – Review the estimated model

5 – Repeat steps 2 and 3 for all models in model space

6 – Compare models



Documents

Dynamic Causal Modelling Advanced Topics SPM Course (fMRI), May 2015 Peter Zeidman Wellcome Trust Centre for Neuroimaging University College London